proove that if a prime number is represented by this phormula p=(2^n)+1 with n>0 then n is a power of 2... it's all for you...
The important fact is that, we can factor:
where is odd.
Now, assume,
is prime where is divisible by odd.
Then,
Thus,
Can be factored as,
It has a proper nontrivial factorization, thus it cannot be prime.
Thus, cannot be divisble by odd number that is,
Thus,
Which were studied by Fermat (my favorite mathemation).